C₇×#15199 #13192415

order	1792
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

direct product

C₇×#15199

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₇	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₇
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ G

1606 conjugacy classes of subgroups; only the 558 normal subgroups are shown.

normal subgroups only

This group has 1606 conjugacy classes of subgroups — too many to draw. Only its 558 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 1792

A minimal faithful permutation representation — 4 generators on 31 points:

(1,2,3,4,5,6,7)(9,11)(13,14)(18,22)(21,23)(24,25,27,29)(26,30,31,28);(12,13,15,14)(17,20)(21,23);(8,9,10,11)(12,14,15,13)(16,17)(18,21)(19,20)(22,23);(16,18,19,22)(17,21,20,23)(24,26,27,31)(25,28,29,30)

Conjugacy classes & character table

448 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 8187);